When \(\mathfrak {g}>0\) , the very definition of \(\nabla _\mathbf {a}W\)  requires some care,
as the variables \((\mathbf {a}, \, \mathbf {d}, \, \xi )\)  are not independent
— they are related to each other by the constraint
\(\xi \in \mathcal {L}(\mathbf {a}, \, \mathbf {d})\) . Our first task is to prove
Lemma REF , which allows us to define \(\nabla _\mathbf {a}W\) 
unambigously. First of all, we recall some results
from [1]} which characterise \(\mathcal {L}(\mathbf {a}, \, \mathbf {d})\) .
We choose closed, simple, geodesic
curves \(\gamma _1\) , ..., \(\gamma _{2\mathfrak {g}}\)  in \(M\) 
whose homology classes generate the first homology group \(H_1(M; \, \mathbb {Z})\) .
(Such curves exist; see e.g. [1]}).
Then, the map
\( \alpha \colon \mathrm {Harm}^1(M)\rightarrow \mathbb {R}^{2\mathfrak {g}}, \qquad \alpha _k(\xi ) := \int _{\gamma _k} \xi \quad \textrm {for } k\in \lbrace 1, \, \ldots , \, 2\mathfrak {g}\rbrace \) 
